This implies that the second turn of millenium was [1‑1‑2001; 00:00:00], exactly one second after [31‑12‑2000; 24:59:59]. As a consequence, the first day of the third millennium was 1‑1‑2001.
Keep in mind that in the framework of our era Thursday 4‑10‑1582 was the very last calendar day of the (proleptic) Julian calendar, and that that Thursday was immediately followed by Friday 15‑10‑1582 being the very first calendar day of the Gregorian calendar. Thus the year 1582 is the only calendar year of our era which had a number of days which is not 365 or 366 (as a matter of fact, it had only 355 days). Between the initial year 1 and the present year 2019 of our era there were only four calendar years of our era whose year number was divisible by 4 but whose number of days was nevertheless 365, namely the years 4, 1700, 1800, and 1900.